We describe the results of an extremely deep , 0.28 deg ^ { 2 } survey for z = 3.1 Ly \alpha emission-line galaxies in the Extended Chandra Deep Field South .
By using a narrow-band 5000 Å filter and complementary broadband photometry from the MUSYC survey , we identify a statistically complete sample of 162 galaxies with monochromatic fluxes brighter than 1.5 \times 10 ^ { -17 } ergs cm ^ { -2 } s ^ { -1 } and observers frame equivalent widths greater than 80 Å .
We show that the equivalent width distribution of these objects follows an exponential with a rest-frame scale length of w _ { 0 } = 76 ^ { +11 } _ { -8 } Å .
In addition , we show that in the emission line , the luminosity function of Ly \alpha galaxies has a faint-end power-law slope of \alpha = -1.49 ^ { +0.45 } _ { -0.34 } , a bright-end cutoff of \log L ^ { * } = 42.64 ^ { +0.26 } _ { -0.15 } , and a space density above our detection thresholds of 1.46 \pm 0.12 \times 10 ^ { -3 } { h _ { 70 } } ^ { 3 } galaxies Mpc ^ { -3 } .
Finally , by comparing the emission-line and continuum properties of the LAEs , we show that the star-formation rates derived from Ly \alpha are \sim 3 times lower than those inferred from the rest-frame UV continuum .
We use this offset to deduce the existence of a small amount of internal extinction within the host galaxies .
This extinction , coupled with the lack of extremely-high equivalent width emitters , argues that these galaxies are not primordial Pop III objects , though they are young and relatively chemically unevolved .